The group of units of ℤ/nℤ

Let $n\geqslant2$ be an integer and $\mathbb{Z}_{n}:=\mathbb{Z}/n\mathbb{Z}$ be the set of all class of integers modulo $n$.

a. Show that for all $a \in (\mathbb{Z}_n^*, \cdot)$, where $\Z_n^*:=\Z_{n}\backslash\{0\}$, such that $\gcd(a,n) = 1$, $a$ has an inverse.

b. Show that for all $a \in (\mathbb{Z}_n^*, \cdot)$ such that $\gcd(a,n) \neq 1$, $a$ does not have an inverse.

c. Let

$$\mathbb{G} = \{a \in \mathbb{Z}_n^* \mid \gcd(a,n) = 1\}$$

Show that $(\mathbb{G}, \cdot)$ is a group, and that it is the largest group contained in $\mathbb{Z}_n^*$.

d. Deduce explicitly the largest group contained in $(\mathbb{Z}_{12}^*, \cdot)$. Give the order and the inverse of each element, and the cardinal of the group. Is this group cyclic?